Question

. Reduce the quadratic form 2x1x2+2x1x3-2x2x3 to a canonicalform by an orthogonal reduction and discuss its nature.​

Answer 1

Answer:

5+7-12

Step-by-step explanation:

2×1×2+2×1×3-2×2×3

5+7-12

12-12

0

Answer 2

Answer:

The canonical form of the quadratic form 2x1x2+2x1x3-2x2x3 is:

2/√3 y1^2 - 2/√2 y2^2 - 2/√2 y3^2 where y1, y2, and y3 are the coordinates in the new orthogonal basis given by the columns of P.

The quadratic form represents a saddle point. In other words, the quadratic form does not have a definite maximum or minimum value, but varies in different directions.

Step-by-step explanation:

To reduce the quadratic form 2x1x2+2x1x3-2x2x3 to a canonical form by an orthogonal reduction, we need to follow the following steps:

1. Create a matrix A with the coefficients of the quadratic form:

A = [0 1 1; 1 0 -1; 1 -1 0]

2. Find the eigenvalues and eigenvectors of A:

The characteristic equation of A is:

det(A - λI) = 0

Where I is the identity matrix and λ is the eigenvalue we are looking for.

Solving this equation, we get:

λ1 = 1, with eigenvector v1 = [1 1 1] / sqrt(3)

λ2 = -1, with eigenvector v2 = [1 -1 0] / sqrt(2)

λ3 = -1, with eigenvector v3 = [0 -1 1] / sqrt(2)

3. Normalize the eigenvectors:

v1 = [1/√3 1/√3 1/√3]

v2 = [1/√2 -1/√2 0]

v3 = [0 -1/√2 1/√2]

4. Construct the orthogonal matrix P with the normalized eigenvectors as its columns:

P = [1/√3 1/√2 0; 1/√3 -1/√2 -1/√2; 1/√3 0 1/√2]

5. Calculate the diagonal matrix D with the eigenvalues on its diagonal:

D = [1 0 0; 0 -1 0; 0 0 -1]

6. Calculate the canonical form by multiplying P^T, D, and P:

P^TDP = [2/√3 0 0; 0 -2/√2 0; 0 0 -2/√2]

Therefore, the canonical form of the quadratic form 2x1x2+2x1x3-2x2x3 is:

2/√3 y1^2 - 2/√2 y2^2 - 2/√2 y3^2

where y1, y2, and y3 are the coordinates in the new orthogonal basis given by the columns of P.

The nature of the quadratic form can be determined by analyzing the signs of the eigenvalues in the diagonal matrix D. In this case, we have one positive eigenvalue and two negative eigenvalues, which means that the quadratic form represents a saddle point. In other words, the quadratic form does not have a definite maximum or minimum value but varies in different directions.

To know more about eigenvectors: https://brainly.in/question/47599349

To know more about saddle point: https://brainly.in/question/33583700

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